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H3 GK IO Gn Hy G G K L Gn Mo Gu Gw Mx Gy Lz G M G M G G L G M G HD G^ G` Hc G H GQ KR LS G M G H G GF Ha G J G L G I  G  J G H G N O N P N N O N O N P N O N P N O N P N N P O N O N P N O N P N N O N  P  N  N P P O N N G GM HV G H G  Q G; GO Hk Rn H H H G Q G H G I G G  J Gj Lm Gn Gr Hy G H G G G5 J8 Gz L G M G L G L G D B @ B G G Q G Q Ge Eg Sh Ej Cw D C D T C D C  D  V ' U I W M U d V`h/77x/:77x/77xH/77xK/)77xI|x FF t @@@ t r t  t@ yy t  P[[MsssG"bx#x$a%xR%xe&A88xh'A/UUxZ+ +? ~? ? ??~2??Simple Harmonic Motion and Energy Conservation using MacMotion Purpose: to investigate Simple Harmonic Motion and Conservation of Energy as applied to a Simple Harmonic Oscillator. Background: Examples of Simple Harmonic Oscillators (SHO) abound in the world around us. Tuning forks and guitar strings undergo simple harmonic motion. Pendula in grandfather clocks and playground swings undergo simple harmonic motion. Electrons in a television tower that broadcasts your favorite show undergo simple harmonic motion. Our prototype for this ubiquitous SHO is a mass m attached to a spring with spring constant k.  If the mass is moved to the right (x > 0) so the spring is extended, there is a force exerted to the left.   If the mass is moved to the left (x < 0) so the spring is compressed, there is a force exerted to the right.   This force that the spring exerts can be written as F = k x where x is the displacement from equilibrium and k is known as the spring constant. For a spring of spring constant k attached to an object of mass m, the period of oscillation is T = 2p (1) As a simple harmonic oscillator moves or oscillates it continuously transforms kinetic energy into potential energy and vice versa. The kinetic energy of a mass m moving with speed v is  (2) The potential energy stored in a spring of spring constant k that has been stretched or compressed a distance x from its equilibrium position is  (3) The total energy, then, is  (4) and, due to conservation of energy, we expect this to remain a constant. At the extremes of motion, when x = xmax = A, the amplitude, the mass momentarily stops so all the energy is due to energy stored in the spring,  (5) As the mass moves through the equilibrium position, when x = 0, the spring has zero potential energy and all the energy is now kinetic energy,  (6) where vmax is the speed of the oscillator as it passes through its equilibrium position. Caution: The springs we use in this experiment are delicate and expensive. Throughout this experiment, be careful that you do not stretch any spring over 40 cm beyond its intial stretch of 100 cm.. If the weight called for or the amplitude called for or anything causes a spring to stretch more than 40 cm, ignore the directions, modify the procedure, tell the instructor. Do not stretch any spring over 40 cm beyond its intial stretch of 100 cm. Procedure: Part 1: Finding the spring constant k While our prototype SHO had a single spring, our actual, experimental SHO consists of a mass m connected between two springs as shown in the sketch. This mass is an air track glider. This means frictional forces may safely be ignored. We need to find the restoring force exerted by the springs as a function of the position of the mass. We will call this the spring constant of the system or the effective spring constant since spring constant usually refers to a single spring and we have two (Think about this for there will be a question about it later).  With the air supply turned on, record the equilibrium position of the glider. Use one edge of the glider as your marker to compare to the distance guide (or ruler) attached to the air track. To determine the spring constant of this system, attach a string to the mass, run the string over the pulley, and attach a 50 g (0.050 kg) weight hanger. Record the new position of the glider. What force are the springs exerting? In increments of 20 g (0.020 kg) increase the mass Mh to 150 g (0.150 kg). But don't overextend the springs! ] Construct a data table including the following:  From this data construct a graph, plotting spring force F vertically and displacement x horizontally. What is the significance of the slope of this line? Part 2: Investigating the period of a simple harmonic oscillator 2a. What determines the period? For starters, does the amplitude affect the period? Use just the bare glider, without additional masses. Determine the period of the SHO by using a stop watch to measure the time for ten periods. Do this for three different amplitudes10 cm, 20 cm, and 30 cm. 2b. Repeat step 2a using using MacMotion and the Motion Detector (the sonic ranging unit) that you have used before. Adjust the time scale so that you take data for more than 10 periods. Measure the time for ten periods and then determine the time of one period. How do your values from 2a and 2b compare? What can you conclude about the effect the amplitude has on the period? If the period is independent of the the amplitude, what does the period depend upon? From theory, we know (or will soon know) the period should be given by T = 2p where m is the mass of the glider (we can easily measure that) and k is the spring constant (How can we determine the spring constant?) The mass of the glider can be varied by adding the small, shiny 50 g masses to the posts on each side of the glider. Starting with the glider itself and then by adding one mass at a time, we can construct SHOs with five different masses. Actually, the springs are moving too, so they also need to be included. But they do not move as much as the glider. To correctly compensate for the mass of the springs, m in the equation T = 2p should really be the mass of the glider plus one-third the mass of the springs; that is m=mglider + (1/3) msprings. Make a data table for the mass and the period. 2c. Determine the period of each SHO by measuring the time for ten periods with a stop watch. Compare this experimental value of the period to the theoretical or expected value of the period, T = 2p. Graphs are always a good way to look at results or to make comparisons. Square both sides of this equation to get T2 = 2 (m/k) or T2 = (2 / k) m. Now make a table and graph T2 vertically (on the y-axis) and m horizontally (on the x-axis). Be sure to include the mass of the springs. What should the slope of this graph be? How close is your value? 2d. Observe the motion of the SHO using MacMotion and the Motion Detector (the sonic ranging unit) that you have used before. Again, compare these experimental values of the period to the theoretical or expected values of the period, T = 2p. As in 2c, do this comparison by plotting T2 vertically and mass m horizontally. Be sure to include the mass of the springs. How close is your value of the slope to the expected value of the slope? Compare your results from 2c and 2d. Part 3: Energy Conservation (This part of the experimen2t requires you to derive an expression for the maximum speed vmax in terms of the amplitude A. This is not optional. This part is important! Understanding this is vital! Don't continue on the basis of someone else's derivation. Thoroughly understand this part yourself.) The total energy of an SHO is E = PE + KE = (1/2) k x2 + (1/2) m v2 E = PEmax = (1/2) k xmax2 = (1/2) A2 = KEmax = (1/2) m vmax2. Place a flag on the glider for better reflection. Use MacMotion to observe the motion of the SHO. Release the glider from some position. Record the position and determine the amplitude (What is amplitude?). Calculate the expected maximum speed vmax as the glider passes through equilibrium. In your lab report, include the expression for vmax in terms of the amplitude A. Determine that speed vmax from the MacMotion graphs. Where is the glider when the speed is maximum? Do this three times for each value of the amplitude. Find the average of the three measurements of the maximum speed vmax and compare the theoretical value with the average of the experimental measurements. Repeat this for three different values of the amplitude10 cm, 20 cm, and 30 cm. Do this for the bare glider without additional masses. Include the mass of the springs so that m = mglider + (1/3) mspring. Part 4: Questions (if time permits): What is the effective spring constant (in terms of k1 and k2) of these arrangements of springs? 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